In solving for the magnitude of the electric field E⃗(z) produced by a sheet charge with charge density σ, use the planar symmet
ry since the charge distribution doesn't change if you slide it in any direction of xy plane parallel to the sheet. Therefore at each point, the electric field is perpendicular to the sheet and must have the same magnitude at any given distance on either side of the sheet. To take advantage of these symmetry properties, use a Gaussian surface in the shape of a cylinder with its axis perpendicular to the sheet of charge, with ends of area A which will cancel out of the expression for E(z) in the end. The result of applying Gauss's law to this situation then gives an expression for E(z) for both z>0 and z<0. (Figure 3) Express E(z) for z>0 in terms of some or all of the variables/constants σ, z, and ϵ0.
1 answer:
Answer:
Check the explanation
Explanation:
Let the charge sheet passes through the middle of the cylinder’s length <em><u>(which is the distance around the end circles)</u></em>, along z-axis, so the cylinder is perpendicular to the surface. Hence the flux through each end will be.....
kindly check the attached image to see the full explanation to the above question.
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Answer:
a)
b) See the proof below
c)
Explanation:
Part a
For this case we have the following differential equation:
With the initial condition
We can rewrite the differential equation like this:
And if we integrate both sides we got:
Where is a constant. If we apply exponential for both sides we got:
Using the initial condition we got:
So then our solution for the differential equation is given by:
For the half life we know that we need to find the value of t for where we have if we use this condition we have:
Applying natural log we have this:
And then the value of t would be:
And using the fact that we have this:
Part b
For this case we need to show that the solution on part a can be written as:
For this case we have the following model:
If we replace the value of k obtained from part a we got:
And we can rewrite this expression like this:
And we can cancel the exponential with the natural log and we have this:
Part c
For this case we want to find the value of t when we have remaining
So we can use the following equation:
Simplifying we got:
We can apply natural log on both sides and we got:
And if we solve for t we got:
We can rewrite this expression like this:
Using properties of natural logs we got: